Theorem pw1m 7534

Description: A truth value which is inhabited is equal to true. This is a variation of pwntru 4312 and pwtrufal 16771. (Contributed by Jim Kingdon, 10-Jan-2026.)

Assertion

Ref	Expression
pw1m	|- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )

Distinct variable group:   x, A

Proof of Theorem pw1m

Step	Hyp	Ref	Expression
1		elpwi 3678	. . . . . . . 8 |- ( A e. ~P 1o -> A C_ 1o )
2		df1o2 6661	. . . . . . . 8 |- 1o = { (/) }
3	1, 2	sseqtrdi 3286	. . . . . . 7 |- ( A e. ~P 1o -> A C_ { (/) } )
4	3	adantr 276	. . . . . 6 |- ( ( A e. ~P 1o /\ x e. A ) -> A C_ { (/) } )
5	1	sselda 3238	. . . . . . . . . 10 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. 1o )
6	5, 2	eleqtrdi 2325	. . . . . . . . 9 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. { (/) } )
7		elsni 3707	. . . . . . . . 9 |- ( x e. { (/) } -> x = (/) )
8	6, 7	syl 14	. . . . . . . 8 |- ( ( A e. ~P 1o /\ x e. A ) -> x = (/) )
9		simpr 110	. . . . . . . 8 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. A )
10	8, 9	eqeltrrd 2310	. . . . . . 7 |- ( ( A e. ~P 1o /\ x e. A ) -> (/) e. A )
11	10	snssd 3839	. . . . . 6 |- ( ( A e. ~P 1o /\ x e. A ) -> { (/) } C_ A )
12	4, 11	eqssd 3255	. . . . 5 |- ( ( A e. ~P 1o /\ x e. A ) -> A = { (/) } )
13	12, 2	eqtr4di 2283	. . . 4 |- ( ( A e. ~P 1o /\ x e. A ) -> A = 1o )
14	13	ex 115	. . 3 |- ( A e. ~P 1o -> ( x e. A -> A = 1o ) )
15	14	exlimdv 1868	. 2 |- ( A e. ~P 1o -> ( E. x x e. A -> A = 1o ) )
16	15	imp 124	1 |- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   C_ wss 3211   (/)c0 3508   ~Pcpw 3669   {csn 3689   1oc1o 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-suc 4492  df-1o 6647

This theorem is referenced by:  pw1if  7535